On Banach spaces with Kasparov and Yu’s property (H). (English) Zbl 1390.46024
G. Kasparov and G.-L. Yu [Geom. Topol. 16, No. 3, 1859–1880 (2012; Zbl 1257.19003)] introduced the following definition: a real separable Banach space \(X\) is said to have property (H) if there exist increasing sequences of finite-dimensional subspaces \(\{X_n\}_{n=1}^\infty\) in \(X\) and \(\{H_n\}_{n=1}^\infty\) in \(\ell_2\) such that the unions of these subspaces are dense in \(X\) and \(\ell_2\), respectively, and there is a uniformly continuous map \(f\) of the unit sphere \(S_X\) onto \(S_{\ell_2}\) such that the restriction of \(f\) to \(S_{X_n}\) is a homeomorphism of \(S_{X_n}\) and \(S_{H_n}\). In the same paper, Kasparov and Yu proved the strong Novikov conjecture for groups which are coarsely embeddable into Banach spaces with property (H).
In this paper, the authors use results of E. Odell and T. Schlumprecht [Acta Math. 173, No. 2, 259–281 (1994; Zbl 0828.46005)] and F. Chaatit [Pac. J. Math. 168, No. 1, 11–31 (1995; Zbl 0823.46016)] on homeomorphisms of spheres to show that Banach lattices with nontrivial cotype have property (H).
In this paper, the authors use results of E. Odell and T. Schlumprecht [Acta Math. 173, No. 2, 259–281 (1994; Zbl 0828.46005)] and F. Chaatit [Pac. J. Math. 168, No. 1, 11–31 (1995; Zbl 0823.46016)] on homeomorphisms of spheres to show that Banach lattices with nontrivial cotype have property (H).
Reviewer: Mikhail Ostrovskii (Queens)
MSC:
| 46B80 | Nonlinear classification of Banach spaces; nonlinear quotients |
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
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